This is a great example of the system failing regular people and also the gross assumption that we all have cars, money and time to spare to drag things around to the recycling centre/dump.
I understand that the cardbord recycling facility is limited by the physical dimensions on the truck that they send around to collect the waste, but they must understand that the alternative is for them to take it away as waste. You shouldn't havr to pay for it to go as bulky waste unless it is really huge, so I would complain to Council Connect about this and ask what method they suggest that you use to recycle the cardboard item.
These are the sorts of everyday issues that we do pay our council tax to see the council resolve. Otherwise oversized recycle-able items will end up in landfill.